If the second two terms on the right hand side were not there, we'd have the result that the latent heat flux is equal to the surface absorbed solar radiation. The two additional terms are positive when Tg < Tsa, and can thus allow the latent heat flux to somewhat exceed the available solar forcing under circumstances when an inversion can form at the surface. The strength of this inversion determines the amount of "excess evaporation" that can be sustained.
We can show that an inversion must always form in the strong evaporation limit, and also put a strict bound on the temperature difference across the inversion. First, note that when Tg = Tsa the last two terms in Eq. 6.38 vanish, while by definition of the strong evaporation limit FL is much greater than the remaining solar term. Thus, when Tg = Tsa, the left hand side exceeds the right hand side. Next, recall that latent heat flux vanishes when the saturation vapor pressure corresponding to the ground temperature equals the vapor pressure at the top of the surface layer. Thus, if the relative humidity is below 100%, FL(Tsa, T) will vanish at some temperature T < Tsa which we will call To. Actually, it is possible that stable boundary layer physics will extinguish the turbulence at some temperature that is somewhat warmer than the temperature at which the gradient of vapor pressure vanishes. In any event, there is a To < Tsa where FL(Tsa,To) = 0. This temperature will be a function of Tsa and the other surface layer parameters, as well as the the thermodynamic properties of the atmosphere. Because the left hand side vanishes at Tg = To, while the right hand side is positive, it follows that the left hand side is less than the right hand side. Together with the previous result, we now know that there is a solution to the surface balance equation with To <Tg < Tsa. The maximun strength of the inversion is Tsa — To, and this determines the maximum excess evaporation, through determining the maximum possible size of the second two terms on the right hand side of Eq. 6.38. It can be shown that Tsa — To increases very slowly with Tsa (Problem ??), whence we conclude that the excess evaporation must increase only slowly with temperature.
In Figure 6.5 we show the latent heat flux as a function of Tsa, obtained by solving Eq. 6.37 for Tg using a simple Newton's method iteration. The calculation was carried out for a water/air atmosphere on Earth with (1 — ag)Sg = 200W/m2, 80% relative humidity in the free atmosphere U = 5m/s and a constant CD = .0015. As expected from our analysis of the weak and strong evaporation limits, the flux grows approximately exponentially at low temperatures, but the growth levels off and becomes much weaker once the latent heat flux exceeds the absorbed
solar radiation. Notably, the latent heat flux grows by more than two orders of magnitude as the air temperature is increased from 280K to 300K, but hardly doubles as it is increased an equal amount to 380K. An examination of the dry Richardson number (proportional to Tsa — Tg) shows that an inversion develops in this circumstance, and becomes stronger as the temperature increases. If water vapor were not positively buoyant, the inversion would become stable, limiting the turbulence and reducing the evaporation even more. However, the buoyancy of water vapor allows the surface layer to remain unstable despite the inversion, especially at high temperatures. In other circumstances permitting a stronger inversion, the surface layer can become stable despite water vapor buoyancy (see Problem ??)
The preceding results shed some light on precipitation rates of water in both cold and warm climates. To turn the latent heat fluxes into precipitation rates, note that 1W/m2 of latent heat flux is equivalent to 1.21 cm/yr of liquid water equivalent precipitation if the flux is due to sublimation, or 1.26 cm/yr if the flux is due to evaporation. It is sometimes erroneously supposed that in the cold conditions of a Snowball Earth, the hydrological cycle shuts down. Let's estimate the precipitation rate for the Snowball tropics. Taking the mean surface absorbed solar flux to be 130W/m2 as is reasonable for ice subject to tropical insolation, and Tsa = 240K, in equilibrium we find that the ice surface temperature is 241.28K. With these temperatures, the latent heat flux is 1.81W/m2, which translates into a precipitation rate of 2.2cm/yr liquid water equivalent. This may not sound like much, but the Snowball can last a very long time. Given 100,000 years to accumulate, this trickle of snowfall can build a glacier 2.2km high, which is high enough to flow significantly. Thus, there is no essential incompatibility with cold Snowball conditions and geological evidence for active glacier flow. The instantaneous noontime precipitation can be much higher, because it is driven by greater solar flux. Only the mean is relevant for building glaciers, but the relatively heavy noontime sublimation, followed by snowfall as night approaches, can be important in modifying the surface conditions and covering dusty, dark ice with fresh, reflective snow during part of the day.
Turning attention next to the warmer conditions of the Earth's present tropical oceans, we take the absorbed solar radiation to be 200 W/m2 and assume Tsa = 300K. Under these circumstances Tg = 301K and the latent heat flux is 125 W/m2, which translates into a precipitation rate of 156 cm/yr. This is reasonably close to the observed tropical precipitation rate. Now suppose that we introduce a high cloud which reflects a lot of sunlight back to space, but which has such a strong greenhouse effect that the change in OLR compensates, leaving the top-of-atmosphere radiation budget unchanged; as we saw in Chapter 5, Earth's actual high tropical clouds do something approximating this idealization. Since the air temperature is determined primarily by the top-of-atmosphere balance in the optically thick limit, we can keep Tsa fixed at 300K as in the previous case. However, the surface absorbed solar will be reduced, say to 100W/m2, while the downwelling infrared is essentially unaffected by the cloud because the tropical atmosphere is optically thick. Under these circumstances, with reduced surface solar flux the surface temperature falls only modestly, to Tg = 299K. However, the latent heat flux falls dramatically, to 57W/m2, by about the same proportion as the reduction in surface solar flux. This example shows that in the optically thick warm regime where the surface is tightly coupled to the air by latent heat exchange, the surface energy budget has little influence on temperature. For the purposes of estimating temperature, we could do pretty well by simply assuming that the ground temperature equals that of the overlying air. However, changing terms in the surface budget - as we did here by reducing the surface solar flux - has a profound effect on precipitation. In brief, in warm tropical conditions, the surface energy budget tells us about precipitation, while the top-of-atmosphere energy budget determines the temperature.
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